University of Illinois Urbana-Champaign

"On the location of zeros of solutions to w"" + Aw = 0 for certain entire functions A"

Steinbart, Enid Marguerite

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Description

Title
"On the location of zeros of solutions to w"" + Aw = 0 for certain entire functions A"
Author(s)
Steinbart, Enid Marguerite
Issue Date
1989
Doctoral Committee Chair(s)
Ting, Tran Wa
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
  • "Consider the differential equation $(\*)$ $w\sp{\prime\prime}$ + $Aw$ = 0 where $A$ is of the form $A$($z$) = $\sum\sbsp{j=1}{m}Q\sb{j}(z)$exp$P\sb{j}(z)$ with $Q\sb{j}$ and $P\sb{j}$ polynomials in $z$. We study the location in the complex plane of the zeros of solutions to $(\*)$. Under mild hypotheses on the $P\sb{j}$'s and $Q\sb{j}$'s, we show that there exist both ""zero-scarce"" regions and ""zero-rich"" regions. We call an unbounded region in the complex plane ""zero-scarce"" if every nontrivial solution of $(\*)$ has only finitely many zeros in the region. In contrast, a region is called ""zero-rich"" if there exists a nontrivial solution of $(\*)$ with infinitely many zeros in the region."
  • We also study the asymptotic behavior of the zeros of nontrivial solutions in zero-rich regions. We show that if $w$ is a nontrivial solution of $(\*)$ with infinitely many zeros in a zero-rich region, then the zeros of $w$ in this region must approach a curve of the form Im$P(z)$ = $K$ where $K$ is a constant and $P$ is a polynomial determined by $A$. We establish a bound for the rate at which the zeros approach this curve.
  • In addition to the classical techniques of ordinary differential equations, we employ methods developed by Walter Strodt. Strodt's general theory examines the existence and asymptotic behavior of solutions to certain first order differential equations in the complex domain.
Type of Resource
text
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http://hdl.handle.net/2142/20092
Copyright and License Information
Copyright 1989 Steinbart, Enid Marguerite

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